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Averages & Means

Generalized Mean / Power Mean Calculator

Explore the power-mean family and its harmonic, geometric, arithmetic, and RMS cases.

Averages & Means

Explore the generalized mean family

p = −1, 0, 1, 2 are named cases
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
Try an example
Positive values

Use ordinary decimals; blank rows are ignored.

Generalized mean

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Understand the subject

What is a generalized mean?

The generalized mean, also called the power mean, is one family that contains several familiar means. Changing one number, p, changes what the calculation emphasizes.

The relationship

The defining relationship

For positive values, p = −1 gives harmonic mean, p = 0 geometric mean, p = 1 arithmetic mean, and p = 2 RMS.

See the idea

One family changes smoothly with p

Worked example

The values 1 and 4

At p = 1, M1 = (1 + 4) ÷ 2 = 2.5. At p = 2, the same values give RMS = √[(1² + 4²) ÷ 2] ≈ 2.915.

Interpret with care

Choose the relationship that fits

Increasing p raises the result for positive unequal inputs, but it does not choose the correct mean for a real task. The quantity’s meaning does that.

For a side-by-side view of the classical means, use the Mean Comparison Calculator.

Quick guide

How to use this calculator

  1. Enter non-negative values.
  2. Choose power p from −100 to 100.
  3. Compare named cases p=−1, 0, 1, and 2 without changing the dataset.

Calculation method

Raise, average, then undo the power

For p≠0, the power mean is the pth root of the mean of xᵖ. At p=0 its continuous limit is the geometric mean. Log-space evaluation avoids unsafe intermediate powers.

Worked example

Recover RMS as a power mean

For 3 and 4, choosing p=2 gives the quadratic mean.

M₂ = √[(3²+4²)/2] = √12.5

Supported inputs

Precision and limits

List size

Add 1–100 labelled value rows.

Decimal size

Each value may contain up to 200 digits and 100 decimal places.

Exactness

Exact rational results are never replaced by a rounded decimal; approximations are labelled.

Domain

Negative values are excluded. A zero with negative p follows the standard generalized-mean zero convention.

Numerical result

Generalized results are approximations; p is limited to avoid a visually different but numerically indistinguishable extremum calculation.